respectively (see Section REF  for relation between the two alternative pictures).
Among the general bilinear deformations, two families are of special interest. The case \(r=s\)  denotes the Castillejo-Dalitz-Dyson (CDD) deformations. These are the deformations that preserve Lorentz invariance. The additional phase factor inherited by the deformed S-matrix is the famous CDD factor. The case \(r=1\)  is also special and shall be called the dynamical hard rod deformation. This family is interesting because it has the nice physical interpretation to correspond to deforming point particles into finite size hard rods. The two families of deformations coincide for \(s=1\) , which is the \(T\bar{T}\)  deformation. Let us comment on the possible values of the index \(s\)  of higher charges \(P_s,H_s\) . A priori, \(s\)  runs over the odd integers. However, for a given theory, in general \(s\)  does not cover all odd values. The possible set of values of \(s\)  is an important characteristic of the model, see [1]} for more details. Since we aim at developing the general framework and do not specify our considerations to a certain theory, we will simply take \(s\)  to be an odd number.
